I'm wondering if the little back room was 'his' room in his daughter's house. The article locates him in Stepney which is where they lived.
Looking at the census, most of the houses on Dempsey Street when Nathaniel was living there look to have been about 7 rooms, so quite large. Nearly all of those I looked at appear to have two families living at the same address, so for example, a family with 5 rooms and a family with 2 rooms, making a total of 7 rooms. There are various combinations such as 6 and 1, so it looks like the householders perhaps rented out a basement or attic room. I am wondering if Nathaniel did this too when living at his daughter's house at 107 Dempsey Street.
The Stangrooms appear to have rented out two rooms in their house after Nathaniel's death. This is pure assumption of course, as we have no idea if he ever lived with his daughter even though he died there.
Dempsey Street is no longer there, but it used to be the next street to the right parallel with Jubilee St (there are flats built in the area and you can see a line of flats along where Dempsey Street used to be

)
[In 1901 the Stangrooms were living at 102 Dempsey St and had a boarder]